A torus is a doughnut-shaped geometric figure formed by revolving a circle in three-dimensional space about an axis coplanar with the circle. To find the surface area of a torus, we can use a straightforward algebraic formula. This article will guide you through the calculation using the formula for the surface area of a torus, provide an example, and demonstrate the final value.
Formula to Calculate the Surface Area of a Torus
The surface area (\( SA \)) of a torus can be calculated using the formula:
\[ SA = 4 \cdot \pi^2 \cdot R \cdot r \]
Where:
- \( SA \) is the surface area of the torus.
- \( R \) is the major radius (distance from the center of the tube to the center of the torus).
- \( r \) is the minor radius (radius of the tube itself).
Explanation of the Surface Area Formula
- Major Radius (\( R \)): This is the distance from the center of the tube to the center of the torus. It essentially represents the larger radius that extends from the center of the torus to the middle of the tube.
- Minor Radius (\( r \)): This is the radius of the tube that forms the torus. It's the smaller radius that represents the cross-sectional circle of the tube.
The formula \( SA = 4 \cdot \pi^2 \cdot R \cdot r \) combines the areas resulting from the circular cross-sections of the torus and the circular path that the tube follows to form the torus.
Example Calculation
Let's use a practical example to illustrate the application of this formula.
Given:
- \( R = 5 \) units (major radius)
- \( r = 2 \) units (minor radius)
We aim to find the surface area of the torus.
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( R = 5 \) units
- \( r = 2 \) units
Step 2: Substitute the Given Values into the Surface Area Formula
\[ SA = 4 \cdot \pi^2 \cdot R \cdot r \]
\[ SA = 4 \cdot \pi^2 \cdot 5 \cdot 2 \]
Step 3: Calculate the Values
\[ SA = 4 \cdot \pi^2 \cdot 10 \]
\[ SA = 40 \cdot \pi^2 \]
Step 4: Calculate the Final Value
Using \( \pi \approx 3.14159 \):
\[ SA \approx 40 \cdot (3.14159)^2 \]
\[ SA \approx 40 \cdot 9.8696 \]
\[ SA \approx 394.78 \]
Final Value
The surface area of a torus with a major radius of 5 units and a minor radius of 2 units is approximately \( 394.78 \) square units.
